In today's world of air and space travel, it has become increasingly important to accurately track an object's in-flight position and velocity. Typically, this is accomplished by one of either angle-only or angle-plus-range tracking methods, depending on the type of sensor used to track the object. Both methods make use of well-known Kalman filtering techniques for estimating the position and velocity of the tracked object. The basics of Kalman filtering are explained adequately in numerous references. By way of example, one such explanation is written by Roger M. Du Plessis in his paper "Poor Man's Explanation of Kalman Filtering", North American Aviation, Inc., June 1967. As explained by Du Plessis, Kalman filtering is an iterative process used to optimize measurement estimates.
In angle-only tracking, sensor measurements are expressed in terms of two angles--elevation and bearing--defined with respect to a computational frame of reference. In other words, the measurements are expressed in a spherical coordinate system. However, the spherical coordinate system has inherent disadvantages with respect to Kalman filters. For example, as the elevation angle approaches .+-.90 degrees, the imaginary line traced by changes in bearing becomes highly curved. This characteristic is similar to that of a circle of constant latitude which becomes highly curved near the north or south pole. The curvature conflicts with the Kalman filter assumption that the measurement process is linear. Furthermore, if the estimated elevation angle is exactly .+-.90 degrees, the Kalman filter cannot process bearing measurements at all since the filter's measurement matrix cannot be properly computed at these angles.
In conventional angle-plus-range tracking, sensor measurements include elevation, bearing and range expressed in the Cartesian coordinate frame. Accordingly, the Kalman filter does not suffer from the aforementioned problems associated with the spherical coordinate representation. However, when the elevation, bearing, and range measurements are expressed in the Cartesian frame, statistical correlation between the three measurement terms are introduced which greatly increases the number of numerical computations required for a filter update.